Math Olympiad | Solving the Exponential Equation {3^m - 2^m = 65}

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[00:00:00]Welcome to my channel. Let's solve this amazing Olympian math question. All right? So, the question says 3 to the power of m - 2 to the power of m is equal to 65.

[00:00:10]So, we are asked to find the value of m.

[00:00:12]This is what to do. It's very simple.

[00:00:14]So, we shall have solution.

[00:00:18]Now, the question says 3 to the power of m - 2 to the power of m equals what? 65.

[00:00:24]Now, first and foremost, um if I introduce one to this equation, look at this. If I write this as 3 to the power of m all to the power of one, nothing change, right?

[00:00:36]- 2 to the power of m all to the power of one is equal to what? 65.

[00:00:42]Nothing change, right?

[00:00:43]Now, let me rewrite this one as 2 over 2 because 2 / 2 is 1.

[00:00:48]So, this become 3 to the power of m all to the power of 2 over 2 because 2 / 2 is 1, right? - 2 to the power of m all to the power of 2 / 2 is equal to what?

[00:01:01]Um 65.

[00:01:04]Now, watch this.

[00:01:05]Let me bring out 1 over 2 from this 2 over 2 and multiply by m. Of course, we shall have 3 to the power of if I bring out 1 over 2 from here and multiply by m, I shall have um m over 2.

[00:01:18]Okay? All to the power of 2, right? - the same thing applicable here. I shall have 2 to the power of m over 2 all what? All squared is equal to what? 65.

[00:01:31]Now, look at this.

[00:01:32]This is 3 to the power of m over 2 and this is 2 to the power of m over 2. So, let me say let let 3 to the power of m over 2 be equal to y.

[00:01:42]Then, let 2 to the power of m over 2 be equal to x.

[00:01:49]So, if I put in this into the given equation, I shall have our y to be what?

[00:01:56]Our y to be Y squared.

[00:01:59]Okay? Minus what?

[00:02:02]X squared.

[00:02:05]Right? Is equal to what? 65.

[00:02:10]But let me re-interchange this to what?

[00:02:12]X.

[00:02:14]And interchange this to what? To Y. I know you see the same.

[00:02:18]This become our X and this become our Y.

[00:02:23]Because always is always the case.

[00:02:27]Now, this is from the difference of two squares, right?

[00:02:33]This 65 can be written as 13 * 5. So, I can write this as X + Y multiplied by our X - Y is equal to what? 13 multiplied by 5 because the the multiple of this will give 65.

[00:02:50]So, I shall form two equations.

[00:02:54]Of course, we can see X + Y will be equal to what? 13, right? And then X - Y will be equal to what? Five. That's equation one and equation two.

[00:03:07]All right. Now, look at this.

[00:03:11]Now, let me multiply or let me add equation one to equation two.

[00:03:15]Okay? So, X + X will give you 2 X of course + Y - um + Y + - Y is zero. That That means that one has gone. This is equal to 13 + 5 is 18.

[00:03:29]Find X by dividing both side by two and divide here by two. This cancel out. X will be equal to 18 / 2 is 9.

[00:03:37]Putting X equal to 9 to any of the equation, let me choose equation one.

[00:03:43]Okay? So, since X is 9, so we shall have 9 + Y is equal to 13.

[00:03:50]By collecting like terms, this means Y is equal to what? 18 minus 9.

[00:03:55]18 minus 9 become what? 4. So, Y is equal to 4. Now, since Y is 4, okay, and our X is 9.

[00:04:04]Let's recall this equation.

[00:04:07]Now, we see that 3 to the power of M over 2 is equal to what? X.

[00:04:15]And our X is 9. So, that means this is what? 9. And 9 can be written as what? 3 squared.

[00:04:21]Okay? So, we shall have Since we have the same base, we shall have M over 2 to be equal to 2 over 1.

[00:04:29]2 * 2 is what? Is 4. So, M will be equal to 4. So, that's the value of M.

[00:04:36]Okay? So, let's test this equation for 2 to the power of M over 2.

[00:04:42]This is 2 to the power of M over 2 is equal to Y.

[00:04:47]And our Y is 4. That means 4. And 4 can be written as what? As 2 to the power of 2. 2 to the power of 2.

[00:04:56]So, the same base, what we are going to do is that our M over 2 is equal to 2. Call this over 1. Cross multiply. 2 * 2 is 4. So, M will be equal to 4, and that happens to be the final answer. Thank you for watching. Share this video. Follow us and subscribe for more math tips like this. Thank you.